Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula

Size: px
Start display at page:

Download "Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula"

Transcription

1 Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula Group 4: Bertan Yilmaz, Richard Oti-Aboagye and Di Liu May, 15

2 Chapter 1 Proving Dynkin s formula We start by defining two crucial building blocks for the proof of Dynkin s formula. Definition 1.1 (Diffusion): A diffusion is a stochastic process X which is the solution to the stochastic differential equation (SDE): dx t = µ(x t )dt + σ(x t )dw t, X = x for t (1.1) where µ, σ and x satisfy the conditions of the existence and uniqueness theorem for SDEs and W t denotes a Wiener process. Definition 1. (Infinitesimal generator): Let X = (X t ) t be a diffusion. The infinitesimal generator L of X t is defined as: E P [f(x t ) f(x) Lf(x) = lim, x R (1.) t + t given that this limit exists. The set of functions f : R n R such that the above limit exists at x is denoted by D A (x), while D A denotes the set of functions for which the limit exists x R n. x + Let X t = X x t now be a 1-dimensional Itô process of the form X x t (ω) = u(s, ω) ds + v(s, ω) db s (ω) where B is a 1-dimensional Brownian motion, x is a constant and f C (R), i.e. f is in C (R) and has compact support in R. Assume that u(t, ω) and v(t, ω) are nice stochastic processes. We moreover assume that u(t, ω) and v(t, ω) are bounded on the set of (t, ω) such that X(t, ω) belongs to the support of f. Define Z = f(x) and apply 1

3 Itô s formula. We obtain: dz = f x (X)dX + 1 f x (X)dX = u f x dt + 1 f x (vdb) + f x (vdb). We have that (vdb) = v (db) = v dt, and thus: f(x t ) = f(x ) + Consequently: E x P[f(X τ ) = f(x)+e x P ( ) u f x + 1 f v x ds + ( ) u f x (X s)+ 1 f v x (X s) ds v f x db. +E x P v f x (X s)db. (1.3) To advance in our proof of Dynkin s formula, we are in the need of defining Borel measurable functions. Definition 1.3 (Borel algebra and Borel measurable function): A Borel algebra B(R) can be defined as B(R) = σ{open sets} where σ denotes a σ-algebra. A real-valued function f : R R which is measurable with respect to the Borel algebra B(R) is said to be Borel measurable. Proposition 1.1: Every continuous function f : R n R is Borel measurable. Let g be a bounded Borel measurable function, i.e. M R such that g M. Then, for all finite T R we have: E x P T g(x s )db s = E x P [ T 1 {s<t} g(x s )db s = since g(x s ) and 1 {s<t} are both F s -measurable. Here, τ T = min(τ, T ) and 1 {s<t} denotes the indicator function defined as: { 1 if s < t 1 {s<t} = if s t.

4 Furthermore, using Itô s symmetry, we obtain: E x P [( τ g(x s )db s τ T g(x s )db s ) = E x P g (X s )ds. E x P [ τ Now we have that: g (X s )ds E x P M ds τ T = E x P[M (τ τ T ) = M E x P[τ τ T T τ T τ T since τ T T τ. Hence: E x P g(x s )db s = lim T Ex P T Combining (1.3) and (1.4) we now obtain: E x P[f(X τ ) = f(x) + E x P g(x s )db s = (1.4) ( ) u f x (X s) + 1 f v x (X s) ds. (1.5) Using (1.5) with τ = t, u = µ and v = σ together with the definition of L in (1.), we arrive at Lemma 1.1. Lemma 1.1: Let X = (X t ) t be a diffusion and the unique solution to the SDE: dx t = µ(x t )dt + σ(x t )dw t, X = x, t. If f C (R), then the limit in (1.) exists and: Lf(x) = µ(x) f(x) x + 1 σ (x) f(x) x. (1.6) We consequently remark that the integrand in (1.5) effectively is the expression for the generator of a diffusion Lf(X s ) with u = µ(x) and v = σ(x). Hence, we finally arrive at Dynkin s formula. Theorem 1.1 (Dynkin s formula): Let f C(R n ) and suppose that τ is a stopping time such that E P [τ <. Then: E x P[f(X τ ) = f(x) + E x P Lf(X s ) ds. (1.7) 3

5 Chapter Applications of Dynkin s formula Here we present a few useful and interesting applications of the powerful Dynkin formula. We start by defining the (first) exit time from an open set. Definition.1 (First exit time): Let D be an open set and X x = (X x t ) t denote a diffusion starting at X = x D. The (first) exit time τ(x) out of D is defined as: τ(x) = inf{t > : X x t / D}..1 The expectation of the exit time from a bounded set In this section, we are aiming to prove that the expectation of the (first) exit time from a bounded set is finite with the aid of Dynkin s formula. We are proving the case for a 1-dimensional domain, but the proof can easily be extended to the multidimensional case. Let X = (X t ) t denote a 1-dimensional diffusion at an interval (a, b), starting at x (a, b) and is thus the unique solution to the SDE: { dx t = µdt + σdw t, t X = x where σ and µ are constants and σ >. 4

6 Integrating from to τ, we first obtain: τ X τ x = µτ + σ dw t where τ denotes the earlier defined (first) exit time. Dynkin s formula reads: E x P[f(X τ ) = f(x) + E x P Lf(X s ) ds. But since we do not know if E[τ <, we cannot use Dynkin s formula directly. Instead, we take τ T for some finite T R and we know that E[τ T <. We obtain: T E P [X τ T = E P [x + µe P [(τ T ) + σe P dw t. Since in our interval (a, b), a < b, we have that: E P [a E P [X τ T E P [b. Since a, b and x are constants, we have that E P [a = a, E P [b = b and E P [x = x. We then obtain: Hence: a x µ T E P dw t =. E P [(τ T ) b x, T >. µ Now, from the monotone convergence theorem, we have that: E P [τ = E P [ lim (τ T ) <. T Hence we conclude that the expectation of the (first) exit time from a bounded set in one dimension is finite. 5

7 . Application 1 We want to show that: J = 1 sdb(s) N (, 1/3) by finding its moment generating function m(u) = E[e uj and using Dynkin s formula. To solve this, we consider the Itô integral: X(t) = sdb(s), t 1 and J = X 1. We have the Itô process X(t) where: dx(t) = tdb(t) and f(x(t), t) = f(x) = e ux. We apply Lemma 1.1 to above and hence obtain: Therefore, by Dynkin s formula: Lf(x) = t u e ux. E[e ux(t) = u s E[e ux(s) ds, where f = and f(x(), ) = 1. Denote h(t) = E[e ux(t). Differentiation with respect to t thus t yields: h (t) = t u h(t) and thus: h (t) h(t) = t u which after integration yields: ln h(t) = 1 u s ds h(t) = e 1 u t3 3 which corresponds to the normal distribution N (, t3 ). Thus, X(t) = 3 sdb(s) N (, t3 3 ) and J = X 1 N (, 1 3 ). 6

8 .3 Application Consider the n-dimensional standard Brownian motion B(t) with B() = and: τ R = inf{t > : B(t) = R}. Find E [τ R. For a n-dimensional standard Brownian motion we have: Lf(B(t)) = 1 f(b(t)). Since it is not known whether E [τ R <, we can not apply Dynkin s formula directly. So we define γ R = τ R T for some finite T R. We thereafter choose f C such that f(x) := x. By Dynkin s formula we now have: [ γr [ γr 1 E [f(b(γ R )) = f()+e f(b(s))ds = +E n ds = n E [γ R. Now from the monotone convergence theorem, we have that: E [τ R = E [ lim T γ R = lim T E [γ R M < for some finite M R. Now, since f(b(τ R )) = B(τ R ) and E [B(τ R ) = R, we finally obtain:.4 Application 3 E [τ R = R n. This example of an application of Dynkin s formula is similar to the previous application but is nevertheless an interesting and highlighting example of how Dynkin s formula can be applied. Consider a flea market in the form of a rectangle with sides a and b, with a mother who has lost her son walking around inside furiously looking for her lost son. We denote the rectangle domain of the market by D and the boundary of the market by Γ = D. The son is in fact waiting for his mother at a point at the boundary of the market, and once his mother reaches any point on the boundary of the market it is just a matter of time before they meet. The question now is, how long time, at most, is it expected that the 7

9 poor son will have to wait for his mother to exit the flea market and find him? Hopefully it is not infinite. Since the woman is hectically and stressfully walking around looking for her son, her movements can be well-described by a -dimensional Brownian motion W t. We consider the searching process to start when the mother is at W = (x, y) R. We define: δ = sup x x Γ, x Γ Γ i.e. the maximum distance from the initial point of the Brownian motion describing the searching process of the mother to the boundary of the rectangular shaped flea market. We realize that δ has to be smaller than the diagonal length of the flea market which is a + b. The generator of a Brownian motion is: L = 1. We now choose a function f C (R ) such that Lf(x, y) = constant and we consequently choose f(x, y) = x + y. Hence: Lf = 1 ( + ) =. But since we do not know if E P [τ < we cannot use Dynkin s formula directly. Instead, we take τ T for some finite T and we know that E P [τ T <. Hence, we obtain from Dynkin s formula: T E P [f(w τ T ) = x + y + E P ds We realize that E P [f(w τ T ) < a + b since E P [W τ < a + b and so:. yielding: x + y + E P [τ T < a + b E P [τ T < a + b x y. We know from the monotone convergence theorem that: Hence: E P [τ = E P [ lim (τ T ) <. T E P [τ < a + b x y a + b 8

10 and the expected time until the mother and her son are reunited is luckily finite. 9

1. Stochastic Processes and filtrations

1. Stochastic Processes and filtrations 1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S

More information

MA8109 Stochastic Processes in Systems Theory Autumn 2013

MA8109 Stochastic Processes in Systems Theory Autumn 2013 Norwegian University of Science and Technology Department of Mathematical Sciences MA819 Stochastic Processes in Systems Theory Autumn 213 1 MA819 Exam 23, problem 3b This is a linear equation of the form

More information

Exercises. T 2T. e ita φ(t)dt.

Exercises. T 2T. e ita φ(t)dt. Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.

More information

Stochastic Calculus. Kevin Sinclair. August 2, 2016

Stochastic Calculus. Kevin Sinclair. August 2, 2016 Stochastic Calculus Kevin Sinclair August, 16 1 Background Suppose we have a Brownian motion W. This is a process, and the value of W at a particular time T (which we write W T ) is a normally distributed

More information

Introduction to Random Diffusions

Introduction to Random Diffusions Introduction to Random Diffusions The main reason to study random diffusions is that this class of processes combines two key features of modern probability theory. On the one hand they are semi-martingales

More information

Bernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012

Bernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012 1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.

More information

Solution for Problem 7.1. We argue by contradiction. If the limit were not infinite, then since τ M (ω) is nondecreasing we would have

Solution for Problem 7.1. We argue by contradiction. If the limit were not infinite, then since τ M (ω) is nondecreasing we would have 362 Problem Hints and Solutions sup g n (ω, t) g(ω, t) sup g(ω, s) g(ω, t) µ n (ω). t T s,t: s t 1/n By the uniform continuity of t g(ω, t) on [, T], one has for each ω that µ n (ω) as n. Two applications

More information

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please

More information

Exercises in stochastic analysis

Exercises in stochastic analysis Exercises in stochastic analysis Franco Flandoli, Mario Maurelli, Dario Trevisan The exercises with a P are those which have been done totally or partially) in the previous lectures; the exercises with

More information

lim n C1/n n := ρ. [f(y) f(x)], y x =1 [f(x) f(y)] [g(x) g(y)]. (x,y) E A E(f, f),

lim n C1/n n := ρ. [f(y) f(x)], y x =1 [f(x) f(y)] [g(x) g(y)]. (x,y) E A E(f, f), 1 Part I Exercise 1.1. Let C n denote the number of self-avoiding random walks starting at the origin in Z of length n. 1. Show that (Hint: Use C n+m C n C m.) lim n C1/n n = inf n C1/n n := ρ.. Show that

More information

Stability of Stochastic Differential Equations

Stability of Stochastic Differential Equations Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010

More information

A Concise Course on Stochastic Partial Differential Equations

A Concise Course on Stochastic Partial Differential Equations A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original

More information

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( ) Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical

More information

p 1 ( Y p dp) 1/p ( X p dp) 1 1 p

p 1 ( Y p dp) 1/p ( X p dp) 1 1 p Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p

More information

Lecture 12: Diffusion Processes and Stochastic Differential Equations

Lecture 12: Diffusion Processes and Stochastic Differential Equations Lecture 12: Diffusion Processes and Stochastic Differential Equations 1. Diffusion Processes 1.1 Definition of a diffusion process 1.2 Examples 2. Stochastic Differential Equations SDE) 2.1 Stochastic

More information

Theoretical Tutorial Session 2

Theoretical Tutorial Session 2 1 / 36 Theoretical Tutorial Session 2 Xiaoming Song Department of Mathematics Drexel University July 27, 216 Outline 2 / 36 Itô s formula Martingale representation theorem Stochastic differential equations

More information

Brownian Motion. 1 Definition Brownian Motion Wiener measure... 3

Brownian Motion. 1 Definition Brownian Motion Wiener measure... 3 Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................

More information

ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary

More information

ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME

ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME SAUL D. JACKA AND ALEKSANDAR MIJATOVIĆ Abstract. We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems

More information

UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE

UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.

More information

Lecture 17 Brownian motion as a Markov process

Lecture 17 Brownian motion as a Markov process Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is

More information

Bernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010

Bernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010 1 Stochastic Calculus Notes Abril 13 th, 1 As we have seen in previous lessons, the stochastic integral with respect to the Brownian motion shows a behavior different from the classical Riemann-Stieltjes

More information

The Wiener Itô Chaos Expansion

The Wiener Itô Chaos Expansion 1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in

More information

Solutions to the Exercises in Stochastic Analysis

Solutions to the Exercises in Stochastic Analysis Solutions to the Exercises in Stochastic Analysis Lecturer: Xue-Mei Li 1 Problem Sheet 1 In these solution I avoid using conditional expectations. But do try to give alternative proofs once we learnt conditional

More information

Brownian Motion on Manifold

Brownian Motion on Manifold Brownian Motion on Manifold QI FENG Purdue University feng71@purdue.edu August 31, 2014 QI FENG (Purdue University) Brownian Motion on Manifold August 31, 2014 1 / 26 Overview 1 Extrinsic construction

More information

The multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications

The multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications

More information

I forgot to mention last time: in the Ito formula for two standard processes, putting

I forgot to mention last time: in the Ito formula for two standard processes, putting I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy

More information

Stochastic Gradient Descent in Continuous Time

Stochastic Gradient Descent in Continuous Time Stochastic Gradient Descent in Continuous Time Justin Sirignano University of Illinois at Urbana Champaign with Konstantinos Spiliopoulos (Boston University) 1 / 27 We consider a diffusion X t X = R m

More information

The concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.

The concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt. The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes

More information

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please

More information

Harmonic Functions and Brownian motion

Harmonic Functions and Brownian motion Harmonic Functions and Brownian motion Steven P. Lalley April 25, 211 1 Dynkin s Formula Denote by W t = (W 1 t, W 2 t,..., W d t ) a standard d dimensional Wiener process on (Ω, F, P ), and let F = (F

More information

Brownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion

Brownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion Brownian Motion An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Background We have already seen that the limiting behavior of a discrete random walk yields a derivation of

More information

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze

More information

Reflected Brownian Motion

Reflected Brownian Motion Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide

More information

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539 Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory

More information

Lecture 12. F o s, (1.1) F t := s>t

Lecture 12. F o s, (1.1) F t := s>t Lecture 12 1 Brownian motion: the Markov property Let C := C(0, ), R) be the space of continuous functions mapping from 0, ) to R, in which a Brownian motion (B t ) t 0 almost surely takes its value. Let

More information

Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation

Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation Jingyi Zhu Department of Mathematics University of Utah zhu@math.utah.edu Collaborator: Marco Avellaneda (Courant

More information

An Introduction to Malliavin calculus and its applications

An Introduction to Malliavin calculus and its applications An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart

More information

Stochastic Areas and Applications in Risk Theory

Stochastic Areas and Applications in Risk Theory Stochastic Areas and Applications in Risk Theory July 16th, 214 Zhenyu Cui Department of Mathematics Brooklyn College, City University of New York Zhenyu Cui 49th Actuarial Research Conference 1 Outline

More information

Stochastic contraction BACS Workshop Chamonix, January 14, 2008

Stochastic contraction BACS Workshop Chamonix, January 14, 2008 Stochastic contraction BACS Workshop Chamonix, January 14, 2008 Q.-C. Pham N. Tabareau J.-J. Slotine Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 1 / 19 Why stochastic contraction?

More information

Verona Course April Lecture 1. Review of probability

Verona Course April Lecture 1. Review of probability Verona Course April 215. Lecture 1. Review of probability Viorel Barbu Al.I. Cuza University of Iaşi and the Romanian Academy A probability space is a triple (Ω, F, P) where Ω is an abstract set, F is

More information

HJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011

HJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011 Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance

More information

Interest Rate Models:

Interest Rate Models: 1/17 Interest Rate Models: from Parametric Statistics to Infinite Dimensional Stochastic Analysis René Carmona Bendheim Center for Finance ORFE & PACM, Princeton University email: rcarmna@princeton.edu

More information

Week 9 Generators, duality, change of measure

Week 9 Generators, duality, change of measure Week 9 Generators, duality, change of measure Jonathan Goodman November 18, 013 1 Generators This section describes a common abstract way to describe many of the differential equations related to Markov

More information

On the submartingale / supermartingale property of diffusions in natural scale

On the submartingale / supermartingale property of diffusions in natural scale On the submartingale / supermartingale property of diffusions in natural scale Alexander Gushchin Mikhail Urusov Mihail Zervos November 13, 214 Abstract Kotani 5 has characterised the martingale property

More information

Stochastic Integration and Continuous Time Models

Stochastic Integration and Continuous Time Models Chapter 3 Stochastic Integration and Continuous Time Models 3.1 Brownian Motion The single most important continuous time process in the construction of financial models is the Brownian motion process.

More information

BROWNIAN MOTION AND LIOUVILLE S THEOREM

BROWNIAN MOTION AND LIOUVILLE S THEOREM BROWNIAN MOTION AND LIOUVILLE S THEOREM CHEN HUI GEORGE TEO Abstract. Probability theory has many deep and surprising connections with the theory of partial differential equations. We explore one such

More information

Harmonic Functions and Brownian Motion in Several Dimensions

Harmonic Functions and Brownian Motion in Several Dimensions Harmonic Functions and Brownian Motion in Several Dimensions Steven P. Lalley October 11, 2016 1 d -Dimensional Brownian Motion Definition 1. A standard d dimensional Brownian motion is an R d valued continuous-time

More information

1. Stochastic Process

1. Stochastic Process HETERGENEITY IN QUANTITATIVE MACROECONOMICS @ TSE OCTOBER 17, 216 STOCHASTIC CALCULUS BASICS SANG YOON (TIM) LEE Very simple notes (need to add references). It is NOT meant to be a substitute for a real

More information

Wiener Measure and Brownian Motion

Wiener Measure and Brownian Motion Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u

More information

Chapter 4. The dominated convergence theorem and applications

Chapter 4. The dominated convergence theorem and applications Chapter 4. The dominated convergence theorem and applications The Monotone Covergence theorem is one of a number of key theorems alllowing one to exchange limits and [Lebesgue] integrals (or derivatives

More information

A Barrier Version of the Russian Option

A Barrier Version of the Russian Option A Barrier Version of the Russian Option L. A. Shepp, A. N. Shiryaev, A. Sulem Rutgers University; shepp@stat.rutgers.edu Steklov Mathematical Institute; shiryaev@mi.ras.ru INRIA- Rocquencourt; agnes.sulem@inria.fr

More information

Some Tools From Stochastic Analysis

Some Tools From Stochastic Analysis W H I T E Some Tools From Stochastic Analysis J. Potthoff Lehrstuhl für Mathematik V Universität Mannheim email: potthoff@math.uni-mannheim.de url: http://ls5.math.uni-mannheim.de To close the file, click

More information

On continuous time contract theory

On continuous time contract theory Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem

More information

A new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009

A new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009 A new approach for investment performance measurement 3rd WCMF, Santa Barbara November 2009 Thaleia Zariphopoulou University of Oxford, Oxford-Man Institute and The University of Texas at Austin 1 Performance

More information

Some Properties of NSFDEs

Some Properties of NSFDEs Chenggui Yuan (Swansea University) Some Properties of NSFDEs 1 / 41 Some Properties of NSFDEs Chenggui Yuan Swansea University Chenggui Yuan (Swansea University) Some Properties of NSFDEs 2 / 41 Outline

More information

1 Introduction. 2 Diffusion equation and central limit theorem. The content of these notes is also covered by chapter 3 section B of [1].

1 Introduction. 2 Diffusion equation and central limit theorem. The content of these notes is also covered by chapter 3 section B of [1]. 1 Introduction The content of these notes is also covered by chapter 3 section B of [1]. Diffusion equation and central limit theorem Consider a sequence {ξ i } i=1 i.i.d. ξ i = d ξ with ξ : Ω { Dx, 0,

More information

Stochastic Differential Equations.

Stochastic Differential Equations. Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)

More information

Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term

Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term 1 Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term Enrico Priola Torino (Italy) Joint work with G. Da Prato, F. Flandoli and M. Röckner Stochastic Processes

More information

Some Terminology and Concepts that We will Use, But Not Emphasize (Section 6.2)

Some Terminology and Concepts that We will Use, But Not Emphasize (Section 6.2) Some Terminology and Concepts that We will Use, But Not Emphasize (Section 6.2) Statistical analysis is based on probability theory. The fundamental object in probability theory is a probability space,

More information

MATH 6605: SUMMARY LECTURE NOTES

MATH 6605: SUMMARY LECTURE NOTES MATH 6605: SUMMARY LECTURE NOTES These notes summarize the lectures on weak convergence of stochastic processes. If you see any typos, please let me know. 1. Construction of Stochastic rocesses A stochastic

More information

STOCHASTIC CALCULUS JASON MILLER AND VITTORIA SILVESTRI

STOCHASTIC CALCULUS JASON MILLER AND VITTORIA SILVESTRI STOCHASTIC CALCULUS JASON MILLER AND VITTORIA SILVESTRI Contents Preface 1 1. Introduction 1 2. Preliminaries 4 3. Local martingales 1 4. The stochastic integral 16 5. Stochastic calculus 36 6. Applications

More information

Feller Processes and Semigroups

Feller Processes and Semigroups Stat25B: Probability Theory (Spring 23) Lecture: 27 Feller Processes and Semigroups Lecturer: Rui Dong Scribe: Rui Dong ruidong@stat.berkeley.edu For convenience, we can have a look at the list of materials

More information

Introduction to Diffusion Processes.

Introduction to Diffusion Processes. Introduction to Diffusion Processes. Arka P. Ghosh Department of Statistics Iowa State University Ames, IA 511-121 apghosh@iastate.edu (515) 294-7851. February 1, 21 Abstract In this section we describe

More information

A Class of Fractional Stochastic Differential Equations

A Class of Fractional Stochastic Differential Equations Vietnam Journal of Mathematics 36:38) 71 79 Vietnam Journal of MATHEMATICS VAST 8 A Class of Fractional Stochastic Differential Equations Nguyen Tien Dung Department of Mathematics, Vietnam National University,

More information

for all f satisfying E[ f(x) ] <.

for all f satisfying E[ f(x) ] <. . Let (Ω, F, P ) be a probability space and D be a sub-σ-algebra of F. An (H, H)-valued random variable X is independent of D if and only if P ({X Γ} D) = P {X Γ}P (D) for all Γ H and D D. Prove that if

More information

Introduction to numerical simulations for Stochastic ODEs

Introduction to numerical simulations for Stochastic ODEs Introduction to numerical simulations for Stochastic ODEs Xingye Kan Illinois Institute of Technology Department of Applied Mathematics Chicago, IL 60616 August 9, 2010 Outline 1 Preliminaries 2 Numerical

More information

MATH4210 Financial Mathematics ( ) Tutorial 7

MATH4210 Financial Mathematics ( ) Tutorial 7 MATH40 Financial Mathematics (05-06) Tutorial 7 Review of some basic Probability: The triple (Ω, F, P) is called a probability space, where Ω denotes the sample space and F is the set of event (σ algebra

More information

Stochastic Integration and Stochastic Differential Equations: a gentle introduction

Stochastic Integration and Stochastic Differential Equations: a gentle introduction Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27 Intro: why Stochastic? Brownian Motion/ Wiener process

More information

Optimal exit strategies for investment projects. 7th AMaMeF and Swissquote Conference

Optimal exit strategies for investment projects. 7th AMaMeF and Swissquote Conference Optimal exit strategies for investment projects Simone Scotti Université Paris Diderot Laboratoire de Probabilité et Modèles Aléatories Joint work with : Etienne Chevalier, Université d Evry Vathana Ly

More information

Stochastic Calculus (Lecture #3)

Stochastic Calculus (Lecture #3) Stochastic Calculus (Lecture #3) Siegfried Hörmann Université libre de Bruxelles (ULB) Spring 2014 Outline of the course 1. Stochastic processes in continuous time. 2. Brownian motion. 3. Itô integral:

More information

Discretization of SDEs: Euler Methods and Beyond

Discretization of SDEs: Euler Methods and Beyond Discretization of SDEs: Euler Methods and Beyond 09-26-2006 / PRisMa 2006 Workshop Outline Introduction 1 Introduction Motivation Stochastic Differential Equations 2 The Time Discretization of SDEs Monte-Carlo

More information

Kolmogorov Equations and Markov Processes

Kolmogorov Equations and Markov Processes Kolmogorov Equations and Markov Processes May 3, 013 1 Transition measures and functions Consider a stochastic process {X(t)} t 0 whose state space is a product of intervals contained in R n. We define

More information

Stochastic integral. Introduction. Ito integral. References. Appendices Stochastic Calculus I. Geneviève Gauthier.

Stochastic integral. Introduction. Ito integral. References. Appendices Stochastic Calculus I. Geneviève Gauthier. Ito 8-646-8 Calculus I Geneviève Gauthier HEC Montréal Riemann Ito The Ito The theories of stochastic and stochastic di erential equations have initially been developed by Kiyosi Ito around 194 (one of

More information

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure

More information

Stochastic optimal control with rough paths

Stochastic optimal control with rough paths Stochastic optimal control with rough paths Paul Gassiat TU Berlin Stochastic processes and their statistics in Finance, Okinawa, October 28, 2013 Joint work with Joscha Diehl and Peter Friz Introduction

More information

Controlled Diffusions and Hamilton-Jacobi Bellman Equations

Controlled Diffusions and Hamilton-Jacobi Bellman Equations Controlled Diffusions and Hamilton-Jacobi Bellman Equations Emo Todorov Applied Mathematics and Computer Science & Engineering University of Washington Winter 2014 Emo Todorov (UW) AMATH/CSE 579, Winter

More information

Elliptic Operators with Unbounded Coefficients

Elliptic Operators with Unbounded Coefficients Elliptic Operators with Unbounded Coefficients Federica Gregorio Universitá degli Studi di Salerno 8th June 2018 joint work with S.E. Boutiah, A. Rhandi, C. Tacelli Motivation Consider the Stochastic Differential

More information

Branching Processes II: Convergence of critical branching to Feller s CSB

Branching Processes II: Convergence of critical branching to Feller s CSB Chapter 4 Branching Processes II: Convergence of critical branching to Feller s CSB Figure 4.1: Feller 4.1 Birth and Death Processes 4.1.1 Linear birth and death processes Branching processes can be studied

More information

Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response

Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response Junyi Tu, Yuncheng You University of South Florida, USA you@mail.usf.edu IMA Workshop in Memory of George R. Sell June 016 Outline

More information

Universal examples. Chapter The Bernoulli process

Universal examples. Chapter The Bernoulli process Chapter 1 Universal examples 1.1 The Bernoulli process First description: Bernoulli random variables Y i for i = 1, 2, 3,... independent with P [Y i = 1] = p and P [Y i = ] = 1 p. Second description: Binomial

More information

ELEMENTS OF PROBABILITY THEORY

ELEMENTS OF PROBABILITY THEORY ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable

More information

Filtrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition

Filtrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition Filtrations, Markov Processes and Martingales Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition David pplebaum Probability and Statistics Department,

More information

Constrained Optimal Stopping Problems

Constrained Optimal Stopping Problems University of Bath SAMBa EPSRC CDT Thesis Formulation Report For the Degree of MRes in Statistical Applied Mathematics Author: Benjamin A. Robinson Supervisor: Alexander M. G. Cox September 9, 016 Abstract

More information

Brownian Motion and Poisson Process

Brownian Motion and Poisson Process and Poisson Process She: What is white noise? He: It is the best model of a totally unpredictable process. She: Are you implying, I am white noise? He: No, it does not exist. Dialogue of an unknown couple.

More information

1 Brownian Local Time

1 Brownian Local Time 1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =

More information

Lecture 4: Introduction to stochastic processes and stochastic calculus

Lecture 4: Introduction to stochastic processes and stochastic calculus Lecture 4: Introduction to stochastic processes and stochastic calculus Cédric Archambeau Centre for Computational Statistics and Machine Learning Department of Computer Science University College London

More information

Weak convergence and Brownian Motion. (telegram style notes) P.J.C. Spreij

Weak convergence and Brownian Motion. (telegram style notes) P.J.C. Spreij Weak convergence and Brownian Motion (telegram style notes) P.J.C. Spreij this version: December 8, 2006 1 The space C[0, ) In this section we summarize some facts concerning the space C[0, ) of real

More information

Particle models for Wasserstein type diffusion

Particle models for Wasserstein type diffusion Particle models for Wasserstein type diffusion Vitalii Konarovskyi University of Leipzig Bonn, 216 Joint work with Max von Renesse konarovskyi@gmail.com Wasserstein diffusion and Varadhan s formula (von

More information

Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form

Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct

More information

Stochastic Differential Equations

Stochastic Differential Equations CHAPTER 1 Stochastic Differential Equations Consider a stochastic process X t satisfying dx t = bt, X t,w t dt + σt, X t,w t dw t. 1.1 Question. 1 Can we obtain the existence and uniqueness theorem for

More information

Stochastic differential equations in neuroscience

Stochastic differential equations in neuroscience Stochastic differential equations in neuroscience Nils Berglund MAPMO, Orléans (CNRS, UMR 6628) http://www.univ-orleans.fr/mapmo/membres/berglund/ Barbara Gentz, Universität Bielefeld Damien Landon, MAPMO-Orléans

More information

Prof. Erhan Bayraktar (University of Michigan)

Prof. Erhan Bayraktar (University of Michigan) September 17, 2012 KAP 414 2:15 PM- 3:15 PM Prof. (University of Michigan) Abstract: We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled

More information

JUSTIN HARTMANN. F n Σ.

JUSTIN HARTMANN. F n Σ. BROWNIAN MOTION JUSTIN HARTMANN Abstract. This paper begins to explore a rigorous introduction to probability theory using ideas from algebra, measure theory, and other areas. We start with a basic explanation

More information

Numerical methods for solving stochastic differential equations

Numerical methods for solving stochastic differential equations Mathematical Communications 4(1999), 251-256 251 Numerical methods for solving stochastic differential equations Rózsa Horváth Bokor Abstract. This paper provides an introduction to stochastic calculus

More information

An Overview of the Martingale Representation Theorem

An Overview of the Martingale Representation Theorem An Overview of the Martingale Representation Theorem Nuno Azevedo CEMAPRE - ISEG - UTL nazevedo@iseg.utl.pt September 3, 21 Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 1 / 25 Background

More information

Sloppy derivations of Ito s formula and the Fokker-Planck equations

Sloppy derivations of Ito s formula and the Fokker-Planck equations Sloppy derivations of Ito s formula and the Fokker-Planck equations P. G. Harrison Department of Computing, Imperial College London South Kensington Campus, London SW7 AZ, UK email: pgh@doc.ic.ac.uk April

More information

OBSTACLE PROBLEMS FOR NONLOCAL OPERATORS: A BRIEF OVERVIEW

OBSTACLE PROBLEMS FOR NONLOCAL OPERATORS: A BRIEF OVERVIEW OBSTACLE PROBLEMS FOR NONLOCAL OPERATORS: A BRIEF OVERVIEW DONATELLA DANIELLI, ARSHAK PETROSYAN, AND CAMELIA A. POP Abstract. In this note, we give a brief overview of obstacle problems for nonlocal operators,

More information

Lecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University

Lecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University Lecture 4: Ito s Stochastic Calculus and SDE Seung Yeal Ha Dept of Mathematical Sciences Seoul National University 1 Preliminaries What is Calculus? Integral, Differentiation. Differentiation 2 Integral

More information

Poisson random measure: motivation

Poisson random measure: motivation : motivation The Lévy measure provides the expected number of jumps by time unit, i.e. in a time interval of the form: [t, t + 1], and of a certain size Example: ν([1, )) is the expected number of jumps

More information